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Theorem ida2 16473
Description: Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i 𝐼 = (Ida𝐶)
idafval.b 𝐵 = (Base‘𝐶)
idafval.c (𝜑𝐶 ∈ Cat)
idafval.1 1 = (Id‘𝐶)
idaval.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ida2 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))

Proof of Theorem ida2
StepHypRef Expression
1 idafval.i . . . 4 𝐼 = (Ida𝐶)
2 idafval.b . . . 4 𝐵 = (Base‘𝐶)
3 idafval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 idafval.1 . . . 4 1 = (Id‘𝐶)
5 idaval.x . . . 4 (𝜑𝑋𝐵)
61, 2, 3, 4, 5idaval 16472 . . 3 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
76fveq2d 6087 . 2 (𝜑 → (2nd ‘(𝐼𝑋)) = (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩))
8 fvex 6093 . . 3 ( 1𝑋) ∈ V
9 ot3rdg 7047 . . 3 (( 1𝑋) ∈ V → (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋))
108, 9ax-mp 5 . 2 (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋)
117, 10syl6eq 2654 1 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3167  cotp 4127  cfv 5785  2nd c2nd 7030  Basecbs 15636  Catccat 16089  Idccid 16090  Idacida 16467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-ot 4128  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-2nd 7032  df-ida 16469
This theorem is referenced by:  arwlid  16486  arwrid  16487
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